CFD Fundamentals in Python: 12 Steps to Navier–Stokes

A hands-on implementation of the numerical methods behind Computational Fluid Dynamics (CFD), progressing from one-dimensional convection equations to solving the incompressible Navier–Stokes equations.

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Overview

This repository documents my journey of learning Computational Fluid Dynamics (CFD) by implementing numerical solvers entirely from scratch in Python.

Rather than relying on commercial CFD software, each exercise develops the governing equations, discretizes them using finite difference methods, and implements the solution using NumPy and Matplotlib.

The exercises begin with simple one-dimensional transport equations and gradually build toward solving the incompressible Navier–Stokes equations.

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Motivation

As an budding researcher in Computational Fluid Dynamics, Thermo-Fluids, and Scientific Machine Learning, I believe understanding the underlying numerical methods is just as important as using existing CFD software.

This repository serves as both:

• a structured learning journal,

• a portfolio of CFD implementations,

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Learning Roadmap

1D Problems

• Step 1 — Linear Convection
• Step 2 — Nonlinear Convection
• CFL Condition
• Step 3 — Diffusion
• Step 4 — Burgers' Equation

2D Problems

• Step 5 — Linear Convection
• Step 6 — Nonlinear Convection
• Step 7 — Diffusion
• Step 8 — Burgers' Equation

Elliptic PDEs

• Step 9 — Laplace Equation
• Step 10 — Poisson Equation

Navier–Stokes

• Step 11 — Lid-Driven Cavity Flow
• Step 12 — Channel Flow

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Numerical Methods Covered

Step	Problem	Dimension	Concepts
1	Linear Convection	1D	Upwind Scheme
2	Nonlinear Convection	1D	Nonlinear PDEs
3	Diffusion	1D	FTCS Scheme
4	Burgers' Equation	1D	Convection + Diffusion
5	Linear Convection	2D	Finite Difference
6	Nonlinear Convection	2D	Coupled PDEs
7	Diffusion	2D	Explicit Time Marching
8	Burgers' Equation	2D	Nonlinear Transport
9	Laplace Equation	2D	Iterative Solvers
10	Poisson Equation	2D	Source Terms
11	Lid-Driven Cavity	2D	Incompressible Navier–Stokes
12	Channel Flow	2D	Pressure-Driven Flow

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Repository Structure

cfd-fundamentals-python/
│
├── README.md
├── LICENSE
├── requirements.txt
├── .gitignore
│
├── notebooks/
│   ├── Step01_Linear_Convection.ipynb
│   ├── Step02_Nonlinear_Convection.ipynb
│   ├── ...
│   └── Step12_Channel_Flow.ipynb
│
├── src/
│   ├── step01_linear_convection.py
│   ├── step02_nonlinear_convection.py
│   ├── ...
│   └── step12_channel_flow.py
│
└── reports/

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Example Results

Results from each step can be seen by opening the corresponding notebook

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Installation

Clone the repository

git clone https://github.com/danielokene/cfd-fundamentals-python.git

Move into the project

cd cfd-fundamentals-python

Install dependencies

pip install -r requirements.txt

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Running a Solver

Example:

python src/step01_linear_convection.py

or open the corresponding Jupyter notebook inside the notebooks/ folder.

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What I'm Learning

Throughout this project I aim to develop a solid understanding of:

• Partial Differential Equations
• Finite Difference Methods
• Explicit Time Integration
• Numerical Stability
• CFL Condition
• Boundary Conditions
• Pressure Poisson Equation
• Incompressible Navier–Stokes Equations
• Scientific Programming with NumPy

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Future Work

After completing these twelve steps, I plan to explore:

• Finite Volume Method (FVM)
• Lattice Boltzmann Method (LBM)
• OpenFOAM
• Turbulence Modeling (RANS & LES)
• GPU-Accelerated CFD
• High-Performance Computing (MPI/OpenMP)

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Acknowledgements

This repository is inspired by the excellent educational project CFD Python: The 12 Steps to Navier–Stokes developed by Prof. Lorena A. Barba and Gilbert F. Forsyth. While this repository contains my own implementations, notes, and learning progress, the learning pathway and pedagogical approach are based on their open educational materials. Please consider citing their work if you use or build upon the original lessons.

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Citation

If you use the original educational material, please cite:

Barba, L. A., & Forsyth, G. F. (2018). CFD Python: the 12 steps to Navier–Stokes equations. Journal of Open Source Education, 1(9), 21. https://doi.org/10.21105/jose.00021

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References

1. Barba, L. A., & Forsyth, G. F. (2018). CFD Python: The 12 Steps to Navier–Stokes. Journal of Open Source Education.
2. Anderson, J. D. Computational Fluid Dynamics: The Basics with Applications.
3. Ferziger, J. H., & Perić, M. Computational Methods for Fluid Dynamics.
4. Versteeg, H. K., & Malalasekera, W. An Introduction to Computational Fluid Dynamics: The Finite Volume Method.

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